How to find coordinates of a triangles vertices

Updated April 24, 2017

By Kelvin Hayes

To find the area of a triangle where you know the x and y coordinates of the three vertices, you'll need to use the coordinate geometry formula: area = the absolute value of Ax(By - Cy) + Bx(Cy - Ay) + Cx(Ay - By) divided by 2. Ax and Ay are the x and y coordinates for the vertex of A. The same applies for the x and y notations of the B and C vertices.

Fill in the numbers for each corresponding letter combination within the formula. For example, if the coordinates of the triangle's vertices are A: (13,14), B: (16, 30) and C: (50, 10), where the first number is the x coordinate and the second is y, fill in your formula like this: 13(30-10) + 16(10-14) + 50(14-30).

Subtract the numbers within the parentheses. In this example, subtracting 10 from 30 = 20, 14 from 10 = -4 and 30 from 14 = -16.

Multiply that result by the number to the left of the parentheses. In this example, multiplying 13 by 20 = 260, 16 by -4 = -64 and 50 by -16 = -800.

Add the three products together. In this example, 260 + (-64) + (-800) to get -604.

Divide the sum of the three products by 2. In this example, -604 / 2 = -302.

Remove the negative sign (-) from the number 302. The area of the triangle is 302, found from the three vertices. Because the formula calls for absolute value, you simply remove the negative sign.

Tips

• To express absolute value, use two vertical lines, one on each side of the formula.

Let D(x1, y1), E(x2, y2) and C(x3, y3) be the mid points of the sides AB, BC and CA of ΔABC. 

Then, the vertices of ΔABC can be found as shown below.

A(x1 + x3 - x2, y1 + y3 - y2)

B(x1 + x2 - x3, y1 + y2 - y3)

C(x2 + x3 - x1, y2 + y3 - y1)

Example 1 :

The mid-points of the sides of a triangle are (5, 1), (3, -5) and (-5, -1). Find the coordinates of the vertices of the triangle. 

Solution :

Let D, E and F be the mid points of the sides AB, BC and CA of ΔABC.

D(x1, y1) = (5, 1)

E(x2, y2) = (3, -5)

F(x3, y3) = (-5, -1)

Vertex A :

A(x1 + x3 - x2, y1 + y3 - y2)

A(5 - 5 - 3, 1 - 1 - (-5))

A(-3, 5)

Vertex B :

B(x1 + x2 - x3, y1 + y2 - y3)

B(5 + 3 - (-5), 1 - 5 - (-1))

B(8 + 5, 1 - 5 + 1)

B(13, -3)

Vertex C :

C(x2 + x3 - x1, y2 + y3 - y1)

C(3 - 5 - 5, -5 - 1 - 1)

C(3 - 10, -5 - 2)

C(-7, -7)

Example 2 :

The mid-points of the sides of a triangle are (5, 3), (4, 0) and (2, 2). Find the coordinates of the vertices of the triangle. 

Solution :

Let D, E and F be the mid points of the sides AB, BC and CA of ΔABC.

D(x1, y1) = (5, 3)

E(x2, y2) = (4, 0)

F(x3, y3) = (2, 2)

Vertex A :

A(x1 + x3 - x2, y1 + y3 - y2)

A(5 + 2 - 4, 3 + 2 - 0)

A(3, 5)

Vertex B :

B(x1 + x2 - x3, y1 + y2 - y3)

B(5 + 4 - 2, 3 + 0 - 2)

B(7, 1)

Vertex C :

C(x2 + x3 - x1, y2 + y3 - y1)

C(4 + 2 - 5, 0 + 2 - 3)

C(1, -1)

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to 

We always appreciate your feedback.

©All rights reserved. onlinemath4all.com

Are you looking to learn how to calculate the vertices of a triangle for math? Do you have a triangle for which you would like to find the vertices? If you answered yes to either of these questions, you are in the right place.

Our triangle vertices calculator will help you find the coordinates of the vertices using the coordinates of the midpoints.

Keep reading to learn:

• What is the vertex of a triangle;
• How to use our triangle vertices calculator; and
• How to find the vertices of a triangle using midpoints.

The point at which two sides of a triangle meet is called a vertex. The word used to refer to more than one vertex is vertices.

To find the triangle's vertices $A$, $B$, and $C$, you need to insert the $x$ and $y$ coordinates of the midpoints $D$, $E$, and $F$, and our calculator will generate the coordinates of the vertices in real-time.

Let's look at the following problem.

A triangle has vertices $A$, $B$, and $C$. The midpoints of the sides labeled $D$, $E$, and $F$ are ($2,3$), ($4,3$), and ($3,1$), respectively. How do we go about finding the vertices using these midpoints?

1. Understand that: $D$ is (x1,y1x_1, y_1x1​,y1​), $E$ is (x2,y2x_2, y_2x2​,y2​), and $F$ is (x3,y3x_3, y_3x3​,y3​)
2. Using the midpoint formula:
   Find vertex $A$:

A=(x1+x3−x2,y1+y3−y2)A = (x_1+ x_3 - x_2, y_1 + y_3-y_2) A=(x1​+x3​−x2​,y1​+y3​−y2​)

Substitute in the values:

$$A=(2+3-4,3+1-3)$$

$$A=(1,1)$$

B=(x1+x2−x3,y1+y2−y3)B = (x_1+x_2-x_3, y_1+y_2-y_3)B=(x1​+x2​−x3​,y1​+y2​−y3​)

Substitute in the values:

$$B=(2+4-3,3+3-1)$$

$$B=(3,5)$$

C=(x2+x3−x1,y2+y3−y1)C = (x_2+x_3-x_1, y_2 + y_3 -y_1)C=(x2​+x3​−x1​,y2​+y3​−y1​)

Substitute in the values:

$$C=(4+3-2,3+1-3)$$

$$C=(5,1)$$

Three. A triangle has three sides and three vertices. The vertices are the points where the three sides of the triangle meet.

To find the vertices of a triangle using the midpoints we use the following steps:

1. Identify the x and y values of the midpoints;
2. Use the midpoint formula: A = (x₁+x₃-x₂, y₁+y₃-y₂); B = (x₁+x₂-x₃, y₁+y₂-y₃); C = (x₂+x₃-x₁, y₂+y₃-y₁);
3. Substitute in the respective x and y values;
4. Calculate.